A button quit on a remote and I popped it open to clean the contacts and found......well, nothing!
There was just a rubber pad with some molded buttons laying on a printed circuit board. At first I thought that maybe the rubber had metal particles mixed in to make it a conductor or magnetized, as the PCB had circuit traces that led together without meeting under each of the elevated buttons.
But a closer look revealed that the pad was not magnetized, nor a conductor and that the PCB had a nice layer of varnish over the hole thing anyway.

Could someone tell me what principle is at work here in the operation of these buttons and what typically goes wrong?

Normally the rubber pads had a coating of a black conductive carbon. The buttons will stop working if the buttons, or contacts are dirty...OR if the conductive coating has Worn off. Dont get too agressive with cleaning or the latter may inadvertantly happen.
If the conductive pad on the back of the rubber keypad measure O/C or excessively high (i.e more than 10 or 20K) then the conductive surface is shot and the best remedy is a new keypad, or remote.
There are a variety of paint on condutors that have been offered as fixes for this, but in my experiance I have found them as much use as a Pocket in your Underpants as they wear off in a matter of weeks, and also clog/bridge the contacts on the PCB.

Carbon impregnated silicon rubber. Clean with a mild detergent and water, don't scrub rinse with clean water, drying with a hair dryer is recommended but hold it in your hand so you don't overheat it. If something water soluable was spilled on the contacts use a paper towel dampened with the same solution to clean those, rinse with another cloth with clean water then clean with residue free contact cleaner, totally denatured alcohol will work, not rubbing alcohol (can have emolients in it).

If it fails to function after cleaning you can try two things. cutting a tiny piece of aluminum (peeled from chewing gum wrapper paper) or copper foil and using contact cement (not stupid glue *note) to hold it in place and see if that fixes it. Or use a conductive nickle paint in as thin a coat as possible. I've tried the silver pens and they suck but I know GC electronics nickel paint has worked for me in the past.http://www.action-electronics.com/gcelect.htm#Paint

One of the more tedious is to find or buy a good/new remote and cut the conductive pad from it and shave cut the one from the broken remote and glue it on. It takes some daring and skill and you could destroy the remote. Again, use contact cement and follow the label instructions

The best is to contact the manufacturer and see if they will sell you just the rubber sheet. Some will but most won't anymore.

*note cyanoacrylate is not flexible and does not take shock well so it disintigrates, it also outgasses and leaves a residue that will coat and kill the other keys.